Overview
This lecture explains how to find the Highest Common Factor (HCF) of two numbers using both lists of factors and prime factorization methods.
Highest Common Factor (HCF) Basics
- HCF is the largest factor shared by two or more numbers.
- To find HCF, list all factors of each number and select the biggest one they have in common.
- Example: Factors of 20 (1, 2, 4, 5, 10, 20) and 28 (1, 2, 4, 7, 14, 28); HCF is 4.
Listing Factors Method
- Write out all factors for each number.
- Identify and choose the largest factor present in both lists.
- Example: Factors of 12 (1, 2, 3, 4, 6, 12) and 18 (1, 2, 3, 6, 9, 18); HCF is 6.
Using Prime Factorization
- Break each number down into its prime factors.
- Identify the prime numbers that appear in both factorizations.
- Multiply all common prime factors (including repeats) to get the HCF.
- Example: Prime factors of 12 (2, 2, 3) and 18 (2, 3, 3); common primes are 2 and 3; HCF is 6.
- If a prime appears more than once in both lists, include it multiple times in the product.
- Example: Prime factors of 12 (2, 2, 3) and 20 (2, 2, 5); common primes are 2 and 2; HCF is 4.
More Prime Factorization Examples
- For 28 (2, 2, 7) and 42 (2, 3, 7), common primes are 2 and 7; HCF is 14.
- For 132 (2, 2, 3, 11) and 420 (2, 2, 3, 5, 7), common primes are 2, 2, and 3; HCF is 12.
Key Terms & Definitions
- Highest Common Factor (HCF) — The largest factor that two or more numbers share.
- Factor — A number that divides another number exactly.
- Prime Factor — A factor that is a prime number (only divisible by 1 and itself).
- Prime Factorization — Breaking a number down into its prime number multipliers.
Action Items / Next Steps
- Practice finding HCF using both the listing factors and prime factorization methods.
- List factors and find HCF for new pairs of numbers for extra practice.